Soil-structure interaction modelling for an offshore wind turbine with monopile foundation

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Master’s Thesis 2016 30stp

Department of Mathematical Sciences and Technology

Soil-structure interaction modelling for an offshore wind turbine with monopile foundation

Steffen Aasen

Environmental physics and renewable energy

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Preface

This thesis constitutes the final work of my master thesis written at NMBU, at the department of mathematical sciences and technology. The work has been done during the spring semester of 2016.

The thesis investigates soil-structure interaction models for offshore wind turbines, with monopile foundation design. It has been contributing to the research project REDWIN, administrated by NGI (Norwegian Geotechnical Institute). Information about the REDWIN project will be given in the introduction chapter for this thesis.

The reader is assumed to be familiar with structural engineering, and especially the finite element method formulation. Analysis within the frequency domain, by use of the fast fourier transform, is also assumed to be familiar to the reader.

Special thanks is given to my main supervisor at NMBU, Tor Anders Nygaard, who has provided helpful and relevant supervision throughout the process of writing this thesis. By opening his network, this thesis has gotten valuable input also from other researchers. I would also like to thank Marit Irene Kvittem at NMBU for giving helpful supervision, and for providing python-scripts that has been used in this thesis.

Special thanks is also given to researchers at NGI, especially Ph.D. candidate Ana Page, who has given valuable help on the geotechnical aspects of the thesis. By taking part in discussing results throughout the project, and giving helpful advice, her contribution has been of high significance. Appreciated feedback and advice has also been given by Kristoffer Skjolden Skau and Jörgen Johansson.

I am also grateful to my co-student Eirik Langeland Knudsen for providing helpful input and discussions throughout the process of this work.

Oslo, May 12, 2016

Steffen Aasen

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Abstract

The last decade, there has been an increased use of offshore sites for harvesting wind energy. With a more complex environment than onshore, leading to higher energy prices, cost reductions becomes important for the industry.

With this in mind, the REDWIN research project has been initiated, supported by the Norwegian Reseach Counsil ENERGIX program. The goal of the program is to reduce cost in offshore wind by integrated structural and geothecnical design. This thesis contributes to this project by investigating the influence on fatigue damage and maximum moments on an OWT, for different soil-structure interaction models. The models has been applied on the NREL 5MW monopile wind turbine.

To investigate the effect of soil damping, three different soil-structure interaction models has been investigated. The reference model has a stiffness matrix at the mudline, according to that developed by Passon (2006) for the Offshore Code Comparison Collaboration 3 (Jonkman and Musial, 2010). The second model uses the same stiffness matrix, with a rotational dashpot damper, to account for soil damping. The third model has been developed by NGI (Norwegian Geotechnical Institute) for the REDWIND project, and is a kinematic hardening soil model.

The accumulated fatigue damage on the OWT structure is reduced by 11% at the mudline, when applying a rotational dashpot damper to account for soil damping. At the tower root, the reduction of fatigue damage is 16%. With the kinematic hardening model, the reduction of fatigue damage is 3% at the mudline, and 7% at the tower root. Both models are seen relative to the reference model, with no soil damping. The results show that damping has a significant effect on fatigue damage for a bottom fixed offshore wind turbine. The difference between the models, are due to different damping characteristics, and implies that a dashpot damper tend to over-estimate soil damping.

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Nomenclature

{𝛷} Modal vector

𝜃_𝑝 Plastic angular displacement (Model C) 𝜈_𝑠 Poisson ratio of the soil

𝜎_𝑖^∗ Slipping stress for kinematic hardening model

C Damping matrix

c Damping coefficient

d Diameter of pile

D Damping factor

D’ Accumulated fatigue damage

E Elastic modulus

Eh Hysteretic loss per load cycle

Ep’ Maximum potential energy for hysteretic loop f* Yield surface function (Model C)

f Load frequency

g Plastic potential function (Model C) G* Equivalent soil shear modulus

Gs Soil shear modulus

H Horizontal force

I Area moment of inertia

K Stiffness matrix

k Stiffness coefficient

m Rate of change of soil shear modulus

M Mass matrix

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VI m’ Negative slope of S-N curve

N Number of cycles before failure p Short for ‘per revolution’ of rotor

Px Axial load on pile

R Radius of foundation footing (Model C)

U Wind speed

u Displacement along inertial x-axis

up Plastic displacement along x-axis (Model C)

Uw Velocity of water

V Vertical force

V0m Peak value of force along x-axis (Model C) W Distributed load along pile

w Displacement along inertial z-axis

wp Plastic displacement along z-axis (Model C)

wpm Peak value of vertical plastic deformation (Model C)

𝜍 Wind shear exponent

𝛾’ Effective unit weight of soil

𝜃 Rotation about y-axis in inertial axis system

𝜔 Angular frequency

𝜙 Angle of friction of soil

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VII

List of acronyms and abbreviations

DNV Det Norske Veritas

EWEA European Wind Energy Association IEA International Energy Agency

IFE Institute for Energy Technology NGI Norwegian Geotechnical Institute NMBU Norwegian University of Life Sciences NREL National Renewable Energy Laboratory

NTNU Norwegian University of Science and Teachnology OC3 Offshore Code Comparison Collaboration 3

OWT Offshore Wind Turbine SWL Still water line

ULS Ultimate limit state

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Inertial axis system in thesis

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List of figures

Figure 1.1: World electricity generation and projection from offshore wind (International Energy

Agency, 2014) ... 1

Figure 1.2: Common designs for bottom fixed OWT’s. Figure after Subhamoy (2014). ... 2

Figure 2.1: Loads on a bottom fixed offshore wind turbine. Figure from www.ovi-lab.be. ... 7

Figure 2.2: Wind speed fluctuation spectrum. Freely after Vorpahl et al. (2012) ... 7

Figure 2.3: Wind profile according to power law. ... 8

Figure 2.4: Spectrum of ocean waves. After Munk (1950). ... 9

Figure 2.5: Jonwap spectrum from Jonkman (2007) ... 10

Figure 3.1: Forces and moments at the mudline ... 12

Figure 3.2: Apparent fixity lenght model ... 12

Figure 3.3: Uncoupled springs model ... 12

Figure 3.4: Example of a reference model of the soil, to make stiffness coefficients. Figure from Page and Skau (2016)... 13

Figure 3.5: P-Y curves for piles. Figure from Reese and Wang (2006). ... 15

Figure 3.6: a) Model C reference system. Figure from Cassidy (1999). b) Spudcan foundation. Figure from Hu et al. (2014) ... 17

Figure 3.7: Yield surface in Model C. Figure from Nguyen-Sy (2005) after Cassidy (1999). ... 18

Figure 3.8: Illustration of the plastic potential function. Figure from Nguyen-Sy (2005) after Cassidy (1999). ... 19

Figure 3.9: Load-displacement curve for: a) p-y curve b) Model with hysteretic behavior c) Kinematic hardening model. ... 20

Figure 3.10: Parallel-series and series-parallel approach. Freely after Iwan (1967). ... 21

Figure 3.11: Hysteretic energy and maximum potential energy, for a single degree of freedom system. ... 22

Figure 4.1: 1p and 3p frequency relative to structure eigen frequencies. ... 23

Figure 4.2: Typical shape of S-N curve... 25

Figure 5.1: Process diagram for working with 3DFloat ... 27

Figure 5.2: Turbulence box in 3DFloat. ... 28

Figure 5.3: Interaction between components in 3DFloat. ... 29

Figure 5.4: Procedure for design verification of 3DFloat model... 30

Figure 5.5: Positions for fatigue damage calculations ... 31

Figure 5.6: Positions around circumference for fatigue calculation a) pile at mudline b) blade root . 32 Figure 5.7: Probability of occurrence for load cases ... 33

Figure 5.8: a) Soil properties according to OC3 Phase II (Jonkman and Musial, 2010) b) Finite element representation of the soil volume and pile. Figure from Page and Skau (2016). ... 34

Figure 5.9: Horizontal displacement at mudline as response to load. Figure from Page and Skau (2016). ... 35

Figure 5.10: Rotation of the pile at mudline as a response to load. Figure from Page and Skau (2016). ... 35

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XII Figure 5.11: Soil-structure interaction models. a) Model 1, b) Model 2, c) Model 3. The figures are

simplified by only showing two dimensions. ... 36

Figure 5.12: Foundation response of Model 3. a) Response from individual spring. b) Total response is given by parallel coupled springs. Figure from Page and Skau (2016). ... 38

Figure 5.13: A set of parallel coupled springs gives a fit to the finite element analysis (FEA) of the soil. From from Page and Skau (2016). ... 39

Figure 6.1: Nacelle in 3DFloat ... 42

Figure 6.2: Visualization of selected mode shapes... 44

Figure 6.3: Eigen frequencies compared with rotor frequency. ... 45

Figure 6.4: Normalized amplitude spectrum of blade root moment (flapwise). ... 45

Figure 6.5: Eigen frequencies according to OC3 Phase II (Jonkman and Musial, 2010). ... 46

Figure 7.1: Free vibration test with 0.1 m tower top displacement. ... 48

Figure 7.2: Free vibration test with 0.1 m tower top displacement, zoomed in. ... 48

Figure 7.3: Fast fourier transform on mudline moments for load case 1. ... 49

Figure 7.4: Fast fourier transform on mudline moments for load case 13. ... 49

Figure 7.5: Total accumulated fatigue damage per year at mudline, arranged by soil-structure interaction model. ... 50

Figure 7.6: Normalized accumulated fatigue damage by load case at mudline ... 51

Figure 7.7: Moment about inertial y-axis at mudline for selected load cases. ... 52

Figure 7.8: Damping ratio provided by Model 3, as a function of load level. Figure from Page and Skau (2016) ... 52

Figure 7.9: Normalized accumulated fatigue damage by load case at mudline, without probability weight. ... 53

Figure 7.10: Maximum moment at mudline arranged after load case. ... 54

Figure 7.11: Total accumulated fatigue damage per year at tower root, arranged by soil-structure interaction model. ... 55

Figure 7.12: Normalized accumulated fatigue damage by load case at tower root. ... 56

Figure 7.13: Maximum moment arranged by load case, at tower root. ... 56

Figure 7.14: Total accumulated fatigue damage per year at tower top, arranged by soil-structure interaction model. ... 57

Figure 7.15: Normalized accumulated fatigue damage at tower top, arranged by load case. ... 58

Figure 7.16: Maximum moment at tower top, arranged by load case. ... 58

Figure 7.17: Total accumulated fatigue damage per year at blade root, arranged by soil-structure interaction model. ... 59

Figure 7.18: Normalized accumulated fatigue damage at blade root, arranged after load case. ... 60

Figure 7.19: Maximum moment at blade root, arranged after load case.. ... 60

Figure 7.20: Free vibration test with new stiffness of Model 3. ... 61

Figure 7.21: Free vibration test with new stiffness of Model 3, zoomed in. ... 62

Figure 7.22: Total fatigue damage at mudline per year, with updated Model 3 stiffness. ... 62

Figure 7.23: Normalized accumulated fatigue damage at mudline after load case, with new Model 3 parameters. ... 63

Figure 7.24: Total fatigue damage at tower root per year, with updated Model 3 stiffness. ... 64

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XIII Figure 7.25: Normalized accumulated fatigue at tower root after load case., with new Model 3

parameters. ... 64

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List of tables

Table 4.1: Frequency of cyclic loads on the NREL 5MW wind turbine ... 23

Table 5.1: Parameters for S-N curves... 31

Table 5.2: Load cases for fatigue damage calculations ... 33

Table 5.3: Parameters for Model 3 ... 39

Table 5.4: Simulations run in 3DFloat ... 40

Table 6.1: Properties of the NREL 5 MW baseline wind turbine. Table from Jonkman and Musial (2010). ... 41

Table 6.2: Tower and monopile defenition in 3D float ... 42

Table 6.3: Eigen frequencies of structure in 3DFloat ... 43

Table 7.1: Sensitivity to soil-structure interaction model according to position on structure. ... 65

Table 7.2: Numerical values for the relative accumulated fatigue damage, according to position on structure. ... 65

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1 Introduction

1.1 Background

Harvesting wind energy has for decades been a big industry in Northern Europe. Denmark has traditionally been a leader, with electricity from wind turbines accounting for roughly 40% of electricity generation (Energinet.dk, 2016). Especially the last decade, European countries has invested largely in wind turbines, with Germany, Great Britain and Denmark being essential (European Wind Energy Association, 2015). With the recent 2020 climate and energy package from the EU, aiming for 20% of energy consumption coming from renewables by 2020, the recent growth in the wind industry is expected to continue.

As visual impact, noise and other factors are introduced in areas with onshore wind farms, there has been a tendency in Northern Europe to look offshore, to further increase the wind energy potential. This increases the complexity of the wind turbines, but its many benefits are driving the industry. Firstly, the meteorological conditions are better for wind harvesting, where the wind speeds are higher and more stable than onshore. Also bigger wind farms can be installed, as larger areas are accessible offshore. The installations so far have mainly been bottom fixed turbines in the North Sea, where several sites are suitable for bottom fixed wind turbines. In 2012 the world energy generation from offshore wind was 15TWh/y, today (2016) it is around 40TWh/y, and by 2020 it is expected to reach 90TWh/y (International Energy Agency, 2014).

Figure 1.1: World electricity generation and projection from offshore wind (International Energy Agency, 2014)

Additional loads are introduced offshore, compared to the onshore environment. Both average- and peak wind loads are higher, and forces from the ocean, mainly waves and currents, further increase loads on the structure. As a response to this, new analytical tools have been developed for OWT’s (Offshore Wind Turbines), to simulate the dynamic behavior of the system under different environmental conditions. Under the OC3 (Offshore Code Comparison Collaboration) and the OC4 project, developed under International Energy Agency Wind, several codes have been compared to verify and validate their accuracy (Robertson et al., 2014). Amongst these codes are 3DFloat, developed by IFE (Norwegian Institute for Energy Technology) and NMBU (Norwegian University of Life Sciences). 3DFloat has mainly been used for floating OWT’s, but as projects involving bottom fixed turbines have become relevant, new developments regarding soil-structure interaction have been initiated. This thesis has been part this process, by testing the influence of different soil models in 3DFloat.

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2 To make offshore wind more competitive in energy markets, the REDWIN project (REDucing cost in offshore WINd by integrated structural and geotechnical design) was initiated. The project brings different professional groups together and aims to develop models that describe soil-structure interaction for offshore wind turbines (Institute, 2016). REDWIN is currently led by NGI, partnering with IFE, amongst others. The 3DFloat code, has so far used crude models such as linear springs and dampers, to model soil-structure interactions. For better modelling, more advanced models are being implemented in the code. This thesis takes part in this process by studying and quantifying the effect on turbine behavior.

1.2 Foundation design in shallow waters

The offshore environment differ largely in soil conditions and sea depth, leading to a variety of possible foundation designs. With material- and installation costs being important for the finances of a wind farm, offshore wind turbines have so far been installed in shallow water, as cost generally grows with increasing depth and distance from shore (Myhr et al.,2014). Common foundation designs for bottom fixed turbines is the monopile, gravity foundation, jacket and tripod.

Figure 1.2: Common designs for bottom fixed OWT’s. Figure after Subhamoy (2014).

Monopiles are preferred in shallow waters, due to its simple technology and low cost. At depths greater than 30m, the jacket and tripod is regarded as suitable designs, as the monopile foundation meet its limitations. For sites with rocky ground, the gravity base can be a suitable foundation, due to difficulties in interfacing the other designs.

The main function of the foundation is to bring the necessary support for the tower. With a mass of roughly 700 tonnes of the tower-nacelle-hub-rotor system (5MW NREL Turbine), a strong vertical support is required. In extreme weather, the horizontal forces from waves and wind give moments in the order of 10⁸ MN at the seabed, creating a need for firm horizontal support from the foundation. With a desired lifetime of roughly 20 years, high standards for the foundation design is necessary.

In this thesis, the monopile foundation has been of interest, as this is the most widespread design, and that of current interest in the REDWIN project.

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1.3 Previous research

Several studies have been conducted regarding soil-structure interaction for monopile foundations. The ones regarded most relevant for this thesis, is briefly be presented below.

Zaaijer (2006) studied the influence of different linear soil-structure interaction models for a 3MW OWT, with a monopile foundation. This was quantified by the influence on the first- and second natural frequencies of the structure. He found that an inaccuracy of 4% on the first natural frequency could be expected (with respect to a finite element representation of the soil), with the best preforming linear model he tested. The foundation models were also compared with experimental results from real offshore sites, to study how the models reproduced experimental results. Five of the sites he studied corresponded with expectations, but two of the sites gave large unexplained deviations.

Carswell et al. (2014) and Carswell et al. (2015) added damping to the mudline response. Firstly by a horizontal dashpot, and secondly by a rotational dashpot at the mudline. In these studies, the NREL 5MW monopile was used. The results were quantified by average and maximum load- and displacement responses at the mudline. For extreme storm conditions, Carswell et al. (2015) found that the maximum and standard deviation of mudline moment was decreased by 7-9%, with mudline damping introduced to the system.

Yung et al. (2015), compared the p-y curve approach (see Chapter 3.2.1) to a finite element representation of the soil-foundation system, for the NREL 5MW monopile. For the mudline moments, the differences was insignificant, but for the tilt angle at the pile head, the difference was significant (>14%).

Damsgaard et al. (2015) studied the effect of varied soil properties, on fatigue damage equivalent moments (FDEM), on a parked 5MW wind turbine with a monopile foundation. He found that stiffness and damping characteristics significantly affected FDEM for the parked wind turbine. A 50% reduction of the deterministic properties gave a 12% increase in FDEM for the stiffness reduction, and 21% for the damping reduction.

Beuckelaers (2015) introduced a different approach for simulating soil damping, by using a kinematic hardening soil model (see Chapter 3.2.3). By simulating a rotor stop for an OWT with a monopile foundation, he compared results with the p-y curve approach for pile analysis, used by DNV and API standards. It was concluded that the kinematic hardening model, forms a suitable basis for time domain calculations for OWT’s.

The studies mentioned above have covered specific loading conditions, which says little of the impact on the OWT in a lifetime perspective. In addition to this, influence on the structure above mudline, has not been given much attention.

With the studies mentioned above as a background, this thesis has used similar soil-structure interaction models, but with a wider range of environmental conditions, and quantification of the impact on fatigue damage and maximum loads, at several locations on the OWT structure. This will give a better understanding of the impact of different soil-structure interaction models in a lifetime perspective.

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1.4 Project description and goals

The main goal of this thesis is to study the influence of different soil-structure interaction models on an offshore wind turbine with a monopile foundation. It is a continuation of a semester thesis by the subject author, where an extreme loading case was studied for three different soil models. A weakness of this study, was that the model of main interest, was calibrated outside the load region of main interest. In addition to this, wind and wave conditions was simplified, and few influences on turbine dynamics was investigated. This thesis introduces additional soil-structure interaction models, a new parameterization of current models, a larger range of environmental conditions, and more results for structure influences.

In addition to supervision from Tor Anders Nygaard (main developer of 3Dfloat), frequent contact with NGI has been held, mainly through the REDWIN Phd. Candidate, Ana Page. Special interest on effects from soil damping, has been expressed by geotechnical engineers at NGI. Based on this, effects of soil damping has been of high interest in the thesis. It should be noted that special attention to soil damping came after the literature study on soil-structure interaction models, which the theoretical chapters in this thesis are influenced by.

Based on the description above the thesis has the following goals:

 Preform a literature study on soil-structure interaction models, and present relevant models for OWT modelling.

 Implementation and verification of the NREL 5MW Monopile design in 3DFloat.

 Study the effects of different soil models on the NREL 5MW monopile wind turbine, with a special focus on soil damping.

To quantify the effects of different soil-structure interaction models, maximum loads and fatigue damage is studied, with fatigue damage being that of main interest. Maximum loads are important especially in connection with permanent deformation of the soil, a critical factor to consider in the design phase.

Fatigue damage is important for potential costs reduction in the OWT industry. If better soil modelling can improve fatigue damage calculations for OWT’s, this has the potential for cost reduction through material savings.

The soil-structure interaction models applied in this thesis are so-called macro element models. This means that the full response of the soil-foundation system, is given at a single point of the OWT structure, normally at the mudline. This approach I widely used in development of soil-structure interaction models. Still, this is contrary to the industry-standard for pile analysis, where the soil response is distributed over the length of the pile (see Chapter 3.2.1). As soil damping is most easily implemented by a macro element model, this approach has been used in this thesis. This is also beneficial with regard to computational burden. This will be further discussed in Chapter 0.

A comparison with the industry standard (p-y curve approach) is planned for, as a continuation of this thesis.

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1.5 Structure of thesis

The thesis has a theoretical part, a methodological part, a presentation of results, a discussion and a concluding part. The chapters do not strictly follow this pattern, therefore a brief introduction of the structure of the thesis is given, to better navigate in the text.

Theoretical part

Chapter 2 gives a general presentation of loads in the offshore environment, with a focus on the frequency domain, as this is of high interest in OWT analysis.

Chapter 3 gives a presentation of soil-structure interaction models for OWT’s. It gives a general overview of important models, and is a result from a literature study on soil-structure interaction models for OWT’s. Only parts of this theory is applied in the thesis.

Chapter 4 presents a theoretical aspect on eigen frequencies and fatigue damage calculations, which will both be used to quantify the influence of different soil-structure interaction models.

Methodological part

Chapter 5 presents the methodologies used in this thesis. This includes:

- A general presentation of the software 3DFloat.

- A presentation of the methodology for verifying the 3DFloat OWT model.

- Specifications regarding fatigue damage calculations.

- Presentation of load cases used in the thesis.

- Presentation of soil profile and soil-structure interaction models investigated in this thesis.

- Presentation of simulations run in 3DFloat Verification of OWT model

Chapter 6 presents the verification process of the NREL 5MW wind turbine in 3DFloat. It will be tested if the 3DFloat model gives reasonable results, compared with other studies on the same structure, before the model is used for further study.

Results and discussion

Chapter 7 presents influences of different soil-structure interaction models, divided in four subchapters:

- Stiffness and damping characteristics of soil-structure interaction models, from free vibration tests.

- Influence of fatigue damage and maximum moments at selected positions on the OWT.

- Influence on fatigue damage with updated stiffness for one of the soil-structure interaction models.

- Summary of results.

Chapter 8 presents the main findings of the thesis. It also suggests improvements in the methodology used in this thesis, and further studies that that would be a natural continuation of this work.

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2 Loads in the offshore environment

As the wind industry has expanded from mainly onshore wind farms, to also include offshore sites, environmental loads on the structure has increased in complexity. With higher wind loads, and hydrodynamic forces introduced, a good description of the subject loads is essential for accurate simulations. In addition to gravitational-, aerodynamic- and hydrodynamic loads, other factors as sea spray, ice impacts, marine growth, scouring and earthquakes influence an OWT (Figure 2.1). As wind and waves are most influential, these will be the focus of this chapter.

As soil-structure interaction is the focus of this thesis, only a brief presentation of aero- and hydrodynamic loads will be presented. The aim is to give the reader a general understanding of these topics, with special attention on the frequency domain. This is of high interest in OWT dynamics, as it is a rotating system, very sensitive to resonance effects.

2.1 Aerodynamic loads

In the frequency spectrum of wind fluctuations, there are mainly three peaks (Figure 2.2). Firstly “The synoptic peak” at around 4 days, due to changing weather, secondly “The diurnal peak” at 12 hours, due to day-/night fluctuations, and thirdly is “The turbulent peak”, at around 1.5 minutes, caused by random fluctuations in wind speed. In structural analysis for monopiles, load periods under five second are of main interest, as this is where the eigen frequencies of the structure is. Load fluctuations in this region, has the potential to cause resonance effects, which greatly influence fatigue life of the structure.

The aerodynamic loads in the frequency region of a few seconds, is mainly caused by aerodynamic effects on the rotor. Two load cycles of high interest is the 1p and 3p load frequency, with p meaning

‘per revolution’. 1p and 3p load frequencies are mainly caused by the wind profile at the site, and tower shadowing effects. These effects are briefly described below.

Figure 2.1: Loads on a bottom fixed offshore wind turbine. Figure from www.ovi-lab.be.

Figure 2.2: Wind speed fluctuation spectrum. Freely after Vorpahl et al.

(2012)

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8 The general wind profile

Due to friction at the earth’s surface, wind speed tends to decrease close to the ground (or sea surface).

As an approximation, a vertical power shear law can be used, to express wind speed as a function of the height:

𝑈(𝑧) = 𝑈_𝑟𝑒𝑓( 𝑧 𝑧_𝑟𝑒𝑓)

𝜍

(2.1) , where U is the horizontal wind velocity at height z, 𝑈_𝑟𝑒𝑓 is the wind speed at some reference height, 𝑧_𝑟𝑒𝑓, and 𝜍 is the shear exponent. This describes an increasing wind speed with height. This feature of the wind should be carefully considered in wind turbine design, as it introduces cyclic loads on the structure. With a rotational frequency, 1p, of the turbine, each rotor blade will experience cyclic loads with this frequency, as it passes maximum height once per revolution. An additional load frequency of 3p is introduced on the structure, when the turbine has three rotor blades, as blades passes the top three times per revolution. Figure 2.3 gives a plot of the wind profile with a wind speed of 12 m/s, a reference height of 90 m, and a shear exponent of 0.14. As seen in the figure, the blade tip speed varies with roughly 2 m/s for a turbine with hub at 90 m, and a rotor diameter of 120 m.

Tower shadowing

The 3p excitation from the blades, is further increased by an effect known as tower shadowing. In front of the tower, wind speeds are lower due to aerodynamic effects caused by the tower. This leads to a decrease in wind speed each time a blade passes the tower, significantly increasing 1p effect on each blade, and the 3p effect on the full structure.

Time domain variations in wind speed

Wind speed will fluctuate in both magnitude and direction. For an overview of wind characteristics in the time domain, the author suggests Vorpahl el al. (2012) for an overview. Long-term variations in wind speed, and turbulence intensity used in this thesis, is presented in Chapter 5.4.

Figure 2.3: Wind profile according to power law.

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2.2 Hydrodynamic loads

The hydrodynamic forces on a bottom fixed OWT is mainly due to forces from waves and currents.

Water moving relative to an immersed body, impose lift- and drag forces on the body, depending on its shape. For a cylinder with water moving perpendicular to its central axis, the lift force is approximately zero. If the water has an acceleration relative to the cylinder, there are additional inertia forces.

Morison’s equation gives an approximation for the forces on a body in a flow of water, and is for a cylinder given by:

𝐹̂ =1

2𝐶_𝑑𝜌𝐷|𝑈_𝑤|𝑈_𝑤+ 𝐶_𝑚𝜌𝐴𝑈_𝑤̇ (2.2)

, where 𝐹̂is force per unit length of the member, Cd is the drag coefficient of the member, 𝜌is the density of water, D is the diameter of the member, Uw is the velocity of the water, Cm is the inertia coefficient, and A is the cross sectional area of the cylinder.

When both the body and fluid is moving relative to a reference point, a relative form of the Morison’s equation can be used. For more details on this, the reader is referred to Jonkman (2007). For hydrodynamic forces on wet elements, 3DFloat uses the relative form of Morison’s equation.

Waves

Figure 2.4 shows a tentative classification of ocean waves, according to wave period. Ordinary gravity waves are of highest significance, as they have periods of a few seconds, which is in the region of the 1^st tower eigen frequencies. The most interesting frequencies for the NREL 5MW Monopile, is around 0,25 hz, or a wave period of 4s, as this is the 1^st tower eigen frequencies (see Chapter 6.2). Even though this is within the spectrum of ordinary gravity waves, the majority of waves considered in this thesis has wave periods over 6 seconds, or below 0.17 Hz, leaving most of them outside the resonance zone.

Figure 2.4: Spectrum of ocean waves. After Munk (1950).

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10 Constant wind over a significant time-span make ordinary gravity waves. Wave height and direction is a function of the wind velocity, which over time changes in magnitude and direction. This leads to intersecting wave systems. Together with this, effects of the cost line leads to an irregular wave surface at offshore sites. A significant factor influencing the wave height and shape is sea depth. At deep sites far from shore, waves tends to be relatively high, with long periods. At intermediate and shallow water sites, waves are smaller, with steeper wave crest, and shorter wave periods.

Several modeling techniques for waves exist, with linear airy theory being a widespread approach for deep water sites. Airy wave theory, models the sea surface with a sinusoidal shape. For shallow waters, where wave crests often are steeper, higher order models as for example Stream function wave theory gives more representative wave modelling. The reader is referred to external sources for more theory concerning this.

To simulate irregular sea states at an offshore site, the Jonswap spectrum is widely used. The Jonswap spectrum gives the wave energy distribution as a function of frequency, as shown in Figure 2.5, here with peak wave frequency at 0.5 rad/s, equivalent to a wave period of 13 s. The gamma factor is used to calibrate the peakedness of the function. An irregular sea state can then be modelled by superposition of regular waves, with wave frequencies according to the Jonswap spectrum. In 3DFloat, an irregualar sea state is made from the superposition of airy waves, with frequencies according to the Jonswap spectrum. Superposition of airy waves, generated from the Jonswap spectrum, will be used in this thesis.

Figure 2.5: Jonwap spectrum from Jonkman (2007)

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3 Soil-structure interaction models

Several mechanisms work together to give the loading response from the soil on an offshore foundation.

The simplest approach assumes a linear response from the soil, where responding forces from the foundation is linearly dependent on displacement of the foundation. The soil stiffness can be determined from for example monotonic load tests. For small displacements, the soil response can be approximated by a linear model, but for larger displacements, nonlinearities and possibly soil deformations, can be significant. Linear models also neglect soil damping effects, which also impact the dynamics of the system. In addition to this, soil properties can change over time due to for example scour (erosion of soil due to flowing water).

In this chapter, different soil-structure interaction models will be presented, as a presentation of a literature study on the topic. The monopile foundation will be the main focus, but also soil-structure interaction models for other offshore foundation structures will be presented, as their mathematical formulations can be translated to the monopile foundation. This presentation will start with the simplest models, formulated by a stiffness matrix at the mudline. Several approaches exist for determining stiffness coefficients, and an overview of these will be given. As nonlinear effects should be considered for more accurate models, especially for high loads, a section with nonlinear models will follow.

As this section is the presentation of a literature study, not all the models is necessary to read, for understanding this thesis. Especially Model C, and the P-y curve approach in Chapter 3.2 has little relevance for understanding the results presented in this thesis. Mathematical formulations of Chapter 3.2 is also of higher complexity, therefore the reader should be familiar with soil-structure interaction formulations.

3.1 Linear foundation models

The models presented in this section are from the studies by Zaaijer (2006) and Passon (2006), on foundation models for offshore wind turbines. The models are two-dimensional, and take only one of the horizontal dimensions into consideration. In an OWT perspective, this means that side-to-side movement of the tower is not considered. Linear models have an advantage, due to their low computational burden.

This class of models is expressed by a stiffness matrix at the mudline. The stiffness matrix represents the response from the soil, given at a single point at the mudline. The response from the soil, F, given at the mudline will be approximated by:

𝐹 = [H M V

] = [

k_uu k_uθ 0 k_θu k_θθ 0

0 0 k_w

] [ u θ

w] (3.1)

, where u is horizontal displacement, w is vertical displacement, and θ is the angular displacement. V, M, and H is their respective forces and moments. The different k’s are corresponding stiffness coefficients. The stiffness matrix approach is easily implemented into a finite element analysis program, as the stiffness matrix can be added to the node at the mudline. Finding the stiffness coefficients for equation (3.1) has several approaches, and will be presented next.

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12

Figure 3.1: Forces and moments at the mudline

Figure 3.2: Apparent fixity lenght model

Figure 3.3: Uncoupled springs model

3.1.1 Stiffness matrix by effective fixity length

This model replaces the soil with an extended pile length, constrained at a length, l, under the mudline as shown in Figure 3.2. The extended pile length will in this way represent the soil response, as the pile is free to move at the mudline node. A pile penetration of four times the pile diameter, gives the best results in the studies by (Zaaijer, 2006). Other papers suggests a pile length from 3.3 to 8 times the pile diameter, depending on soil conditions (Kühn, 1995). With a given pile length, the stiffness matrix at the mudline can be expressed by:

𝑲 =𝟑𝑬𝑰 𝒍^𝟑 [

6 −3𝑙 0

−3𝑙 2𝑙² 0 0 0 𝐴𝑙²

2𝐼

] (3.2)

, with E being the E-modulus of the pile, A being the cross sectional area of the pile, and I being the area moment of inertia of the pile.

3.1.2 Stiffness matrix by uncoupled springs

The stiffness of the soil-pile system, can be expressed as a set of uncoupled springs at the mudline as in Figure 3.3. An uncoupled model, has stiffness coefficients only on the matrix diagonal, meaning that the different degrees of freedom have no impact on each other. Stiffness coefficients of the springs can be determined by static analyses from a reference model, for example a finite element model of the soil- structure system. Given the displacements for a representative load case at the mudline, the spring stiffness’s can be found by solving:

[𝐻 𝑀 𝑉

] = [

𝑘_𝑢𝑢 0 0 0 𝑘_𝜃𝜃 0

0 0 𝑘_𝑤

] [ 𝑢 𝜃

𝑤] (3.3)

, with the stiffness coefficients being the unknowns.

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13 3.1.3 Coupled stiffness matrix from a reference model

In a coupled model, the stiffness matrix has additional coefficients, compared with an uncoupled model.

In this way, the different degrees of freedom have impact on each other. From the studies by Zaaijer (2006), this proves to give a better representation of the soil response. By applying different load combinations on a reference model, a coupled stiffness matrix can be obtained. From two lateral load cases and one axial load case, the stiffness matrix can be determined. A solution of the following equation is suggested by Zaaijer (2006) to obtain the stiffness coefficients:

[ 𝐹₁ 𝑀₁ 𝐹₂ 𝐹₃

] = [

𝑥₁ 𝜃₁ 0 0 0 𝑥₁ 𝜃₁ 0 𝑥₂ 𝜃₂ 0 0 0 0 0 𝑘_𝑧

] [ 𝑘_𝑥𝑥 𝑘_𝑥𝜃 𝑘_𝜃𝜃 𝑘_𝑧

] (3.4)

A full finite element model of the soil volume and the foundation, can for example be used as a reference model, as in Figure 3.4.

Figure 3.4: Example of a reference model of the soil, to make stiffness coefficients. Figure from Page and Skau (2016).

3.1.4 Stiffness matrix from Randolph’s continuum model

This approach builds on the results from analyzing flexible piles response to lateral loading. Randolph (1981) suggested an equivalent shear modulus of the soil to be:

𝐺^∗= 𝐺_𝑠(1 +3𝜈_𝑠

4 ) (3.5)

, where Gs is the shear modulus of the soil and and 𝜈_𝑠 is the Poisson’s ratio of the soil. According to his analyses, a pile in an elastic continuum would behave as an infinite long pile with shear modulus G*, when

𝑙 𝑑≥ (𝐸_𝑝

𝐺^∗)

27 (3.6)

Here l is the length of the pile, d is the outer diameter of the pile, and Ep is the effective Young’s Modulus of the pile. Assuming the vertical degree of freedom is constrained, and with a linearly increasing soil

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14 shear modulus, the elements of the stiffness matrix at the mudline is according to Randolph (1981), given by:

𝑘_𝑢𝑢= 4.52 ∙ 𝑚^∗∙ 𝑟₀²∙ ( 𝐸_𝑝 𝑚^∗𝑟₀)

13

(3.7)

𝑘_𝑢𝜃= 𝑘_𝜃𝑢 = −2.40𝑚^∗𝑟₀³[ 𝐸_𝑝 𝑚^∗𝑟₀]

59

(3.8)

𝑘_𝜃𝜃= 2.16𝑚^∗𝑟₀⁴[ 𝐸_𝑝 𝑚^∗𝑟₀]

79

(3.9)

, where r0 is the radius of the pile, and 𝐸_𝑝= ₁^𝐸𝐼

64𝜋𝐷⁴ and 𝑚^∗= 𝑚 ∙ (1 +³

4𝜈) (3.10)

, with m being the rate of change of the soil shear modulus, which can be obtained from a linear fit to the actual soil properties of the relevant site. The vertical degree of freedom is constrained.

3.1.5 Inclusion of linear soil damping, with a damping matrix

In addition to the stiffness matrix implemented at the mudline, a damping matrix can be added, to account for energy dissipation in the soil. The dynamics at the mudline node, will then be expressed by:

[H M V

] = [

k_uu k_uθ 0 k_θu k_θθ 0

0 0 k_w

] [ u θ w] + [

c_uu c_uθ 0 c_θu c_θθ 0

0 0 c_w] [u̇

θ̇

ẇ

] (3.11)

, where the different c’s are damping coefficients. A procedure for finding representative damping coefficients is presented in Chapter 3.3.

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15

3.2 Nonlinear foundation models

In this section, a selection of nonlinear models for soil-structure interaction will be presented. For small load amplitudes, a linear response can be a good approximation, but for higher loads, nonlinear effects are expected to increases in magnitude. Extreme loads can lead to permanent deformation of the soil, and effects such as stiffening/softening of the soil around the foundation over time, is relevant to account for, as it can influence eigen frequencies of the full OWT structure.

The first model presented, representing the soil with so called p-y curves, is a common approach for analyzing piles in the offshore industry (Pradhan, 2012). An advantage of this approach, is that it models the actual behavior of pile under the mudline. The other models presented are macro element models, where the full response of the foundation is given in a single point at the mudline.

3.2.1 P-Y curves for laterally loaded piles

The p-y curve method is the most common approach for analyzing piles in the offshore industry. It has been developed during several decades, with the contribution from Matlock and Reese (1960), and Matlock (1970), being essential. The method is recommended by API (American Petroleum Institute, 2000) for designing offshore platforms with pile foundations, and by DNV (Det Norske Veritas, 2014) for offshore wind turbine design (mainly for ULS analysis). Due to its widespread usage, several procedures are developed for obtaining p-y curves, both with various soil properties and loading conditions.

The method applies a set of springs along the pile as shown in Figure 3.5. The standard beam column equation is applied to give the response of the pile, with the springs representing the soil resistance. The equation is according to Reese and Wang (2006) given by:

𝑑²

𝑑𝑥²(𝐸_𝑝𝐼_𝑝𝑑²𝑦

𝑑𝑥²) + 𝑃_𝑥(𝑑²𝑦

𝑑𝑥²) − 𝑝(𝑦) − 𝑊 = 0 (3.12)

Figure 3.5: P-Y curves for piles. Figure from Reese and Wang (2006).

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16 , where Px is the axial load on the pile, y is the lateral deflection of pile along the length of the pile, p is the soil resistance per unit length, W is the distributed load along the length of the pile, Ep is the elastic modulus of the pile and IP is the area moment of inertia of the pile. In a finite element analysis of the structure, p-y curves can be attached to nodal points along the pile foundation.

Determining p-y curves

Depending on the soil properties, several methods exist for determining the p-y curves. Semi-empirical solutions has been suggested for elastic springs, with stiffness as a function of depth. Matlock and Reese (1960) gives expressions where the soil stiffness vary with depth in a power- or polynomial form.

DNV (Det Norske Veritas, 2014) suggests semi-empirical methods for determining p-y curves for clay and sand, both for static and cyclic loading. API (American Petroleum Institute, 2000) suggest the same routines.

Alternatively, p-y curves can be extracted from numerical analysis of the soil-structure interaction. A finite element model of the soil volume, as given in Figure 3.4, can be used to obtain p-y curves at different depths.

An experimental test would best represent soil p-y characteristic at the relevant site, as the methods above can vary significantly in the resulting p-y curves. Pradhan (2012) compared p-y curves obtained from API standards, with p-y curves from FEM modelling in the software PLAXIS 3D. For large pile diameters, the two approaches gave significant deviations. Pradhan (2012) argues that this is due to the pile behaving more and more like a rigid pile as the pile diameter increase. API standards are based on piles with relatively small diameter, giving them more flexible properties.

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17 3.2.2 Model C

The following model is primarily developed for shallow offshore foundations, specifically the spudcan foundation (Figure 3.6b). It is included in this thesis as an example of a mathematical formulation that accounts for plasticity effects of the soil, hardening of the soil and yield behavior. With experimental results for a monopile, the parameters of Model C could be fitted to account for monopile behavior.

Model C was developed by Cassidy (1999) for analyzing jack-up structures subjected to random waves, based on a series of loading test performed by Gottardi and Houlsby (1995). It is a macro element model for a spudcan footing on dense sand, where the response from the foundation is given in a single point connected to the superstructure. The model gives vertical-, rotational- and horizontal responses from the soil, as a function of the respective displacements of the footing. Figure 3.6a gives a basic overview of displacements due to applied loads for the foundation.

a) b)

Figure 3.6: a) Model C reference system. Figure f rom Cassidy (1999). b) Spudc an foundation.

Figure from Hu et al. (2014)

The most important mathematical formulations of the model are:

 The yield surface, which defines the plastic and elastic load regions for the model.

 An elastic response function, to represent displacement increments in the elastic load region.

 A flow rule, to represent plastic displacement increments in the plastic load region.

 A hardening rule, defining the size of the yield surface. The size of the yield function is a function of vertical plastic displacement, and does not depend on the horizontal- and rotational displacements.

Briefly summarized the model uses the yield surface to determine whether there is a plastic or elastic response from the foundation. Elastic responses are given by linear coefficients, and plastic responses are given by the flow rule. The size of the yield surface is a function of the vertical plastic deformation, with their relation given by the hardening rule. A description of the yield surface, elastic response function, flow rule and hardening rule will be given below.

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18 Yield surface

A simplified form of the yield surface applied in model C is given by:

𝑓^∗= (ℎ ℎ₀)

2

+ (𝑚

𝑚₀)²− 16𝑣²(1 − 𝑣)² = 0 (3.13) , where ℎ = ^𝐻

𝑉₀, 𝑚 = ^𝑀

2𝑅𝑉₀, 𝑣 = ^𝑉

𝑉₀. R is the radius of the subject footing and Vo is a parameter that defines the size of the yield surface, given by equation (3.16). As shown in Figure 3.7, the yield surface takes the shape of rugby ball. For load cases inside the yield surface, the models gives a linear response to displacements. On the yield surface, the response is given by the flow rule.

Figure 3.7: Yield surface in Model C. Figure from Nguyen-Sy (2005) after Cassidy (1999).

Elastic response function

The relation between incremental loads (dV, dM, dH) and increment displacements (dw,d𝜃,du), is for the elastic case given by:

[ 𝑑𝑉 0.5𝑅⁻¹ 𝑑𝑀

𝑑𝐻

] = 2𝑅𝐺 [

𝑘_𝑣 0 0 0 𝑘_𝑚 𝑘_𝑐 0 𝑘_𝑐 𝑘_ℎ

] [ 𝑑𝑤_𝑒 2𝑅𝑑𝜃_𝑒

𝑑𝑢_𝑒

] (3.14)

, where R is the radius of the footing, G is a representative shear modulus and the different k’s are dimensionless constants.

Flow rule

The flow rule for plasticity in Model C is given by:

[ 𝑑𝑤_𝑝 𝑑𝜃_𝑝 𝑑𝑢_𝑝

] = 𝛬

[ 𝑑𝑔 𝑑𝑉𝑑𝑔

𝑑𝑀𝑑𝑔

𝑑𝐻 ]

(3.15)

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19 , where p denotes plastic displacements, Λ is a non-negative scalar, and g is a plastic potential function, similar to the yield function. Often the yield function, f, would be used in place of g, but to better represent experimental data, the plastic potential function g is introduced. For a detailed description of the plastic potential function, g, the reader is referred to Cassidy (1999). In a geometric interpretation of the flow rule, the direction of the increment displacement vector is normal to the surface of the plastic potential function as in Figure 3.8.

Figure 3.8: Illustration of the plastic potential function. Figure from Nguyen -Sy (2005) after Cassidy (1999).

Hardening rule

The hardening rule defines how the size of the yield surface change, as a function of plastic deformation in the vertical direction, given by wp. The hardening rule is a fit to experimental data, and is given by:

𝑉₀= 𝑘𝑤_𝑝

1 + (𝑘𝑤_𝑝𝑚

𝑉_0𝑚 − 2) ( 𝑤_𝑝

𝑤_𝑝𝑚) + ( 𝑤_𝑝

𝑤_𝑝𝑚)² (3.16)

, where k is an initial plastic stiffness, V0m is the peak value of V0 , and wpm is the corresponding plastic deformation at this peak value. Relevant to note is that the vertical plastic deformation, wp,is the only variable of the function. From Equation (3.13) it can be seen how the yield surface change as the parameter 𝑣, is a function of V0.

Evaluation of the model

The capability of the Model C has been evaluated by Houlsby and Cassidy (2002). This was done by comparing numerical results from the model, to a set of data from tests reported by Gottardi and Houlsby (1995). Results showed high accuracy for the soil conditions of this experiment.

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20 3.2.3 Kinematic hardening model

A kinematic hardening soil model simulates both yield behavior and hysteretic damping from the soil.

By following a different load-displacement curve for loading and unloading, hysteretic damping is introduced to the system. By also allowing the yield point to move, yield behavior is accounted for.

In the p-y curve approach, each p-y curve follows the same elastic curve for loading and unloading. This is illustrated in Figure 3.9a. The p-y curves models the non-linear behavior of the soil, but effects such as damping and yield behavior is ignored.

The soil damping effect also influences the structure, as energy is taken out from the OWT system. The effects of soil damping on a monopile OWT, was studied by Carswell et.al. (2014). Especially in extreme conditions, the damping effect considerately affected the dynamics of the structure. With soil damping included, maximum moment at the mudline was reduced by 7-9% according to the study.

a) b) c)

Figure 3.9: Load-displacement curve for: a) p-y curve b) Model with hysteretic behavior c) Kinematic hardening model.

Figure 3.9b shows a load-displacement curve of a kinematic hardening model as a response to cyclic loading. The area within the hysteresis loops, represents energy absorbed by the soil, which gives a damping effect. In a kinematic hardening model, the yield surface remains the same in shape, but translates in stress space. For a 1D model this can lead to behavior as given in Figure 3.9c.

Mathematical formulation by Iwan (1967)

Motivated to make a mathematical formulation of cyclic systems with hysteretic behavior, Iwan (1967) formulated a class of models to describe the behavior of composite systems. This formulation has in some cases been adapted to OWT soil-structure interaction modelling, as for example in the study by Beuckelaers (2015). A similar approach is also adapted in this thesis (Model 3 – see Chapter 5.6.3).

In this formulation, the Bauschinger effect was essential in the model formulation. The Bausschinger effect states that plastic yield in a material increases the yield strength in direction of plastic flow, and reduces yield strength in the opposite direction. The formulation by Iwan (1967) applies a set of elastic elements, and slip elements to represent the system behavior. The elements are organized in a parallel- series or series-parallel configuration as in Figure 3.10.

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21

Figure 3.10: Parallel-series and series-parallel approach. Freely after Iwan (1967).

In Iwan (1967), each spring element is assigned an representative area Ai, with an elastic modulus Ei,

and critical slipping stress for each slip element, 𝜎_𝑖^∗. The loading behavior of the parallel-series model is then represented by:

𝜎 =𝐹

𝐴= ∑𝐸_𝑖 𝑁𝜖

𝑛 𝑖=1

+ ∑ 𝜎_𝑖^∗ 𝑁

𝑁

𝑖=𝑛+1

, where the summation from 1 to n includes all elements which remain elastic after loading to a strain, 𝜖, and the summation from n+1 to N includes all elements that have slipped or yielded. A similar approach is adapted in this thesis for Model 3 (Chapter 5.6.3).

For the mathematical formulation for the series-parallel model, the reader is referred to Iwan (1967).

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22

3.3 Soil damping formulation by viscous damper

Soil damping in OWT analysis refers to the absorption of energy by the soil, from the OWT system.

There are two main forms of soil damping; hysteretic material damping, and radiation damping. The radiation damping is due to geometric dissipation of waves, and the hysteretic material damping, due to hysteretic effects in the soil. Geometric dissipation is regarded negligible for frequencies under 1Hz, and since this is the main frequency domain for OWT’s, hysteretic material damping is of main interest (Carswell et. al, 2015).

Except from the soil model that includes a viscus damping through a damping matrix, only model C and the kinematic hardening model introduce soil damping. In these models, energy is absorbed by the soil when yield behavior occurs. Chapter 3.1.5 briefly introduces how soil damping can be implemented through a damping matrix. An approach to find a representative damping coefficient is given below.

One of the soil models in this thesis introduces damping based on this approach (see Model 2 – Chapter 5.6.2).

For a soil-structure interaction model to represent soil damping, the right amount of energy should be absorbed by the soil for each load cycle. The hysteretic loss per cycle is has the notation Eh. For a single degree of freedom system, with a linear spring and a dashpot damper, the hysteretic curve can look as in Figure 3.11, with the hysteretic energy loss being the area inside the hysteretic loop. The maximum potential energy is has the notation Ep’. The damping provided by the system can be quantified by the damping factor D, given by:

𝐷 = 1 4𝜋∙𝐸_ℎ

𝐸_𝑝^′ (3.17)

Knowing the hysteretic damping energy loss of a system, an equivalent horizontal (or rotational) viscous damper can be introduced, representing the damping of the system. The viscous damping coefficient can according to Chopra (2007) be represented by:

𝑐 = 𝐸_ℎ 2𝑢²𝜋²𝑓

(3.18)

, where u is the horizontal displacement of the system, and f is the load frequency. Following this, the damping value will be representative for a given frequency and displacement level.

Figure 3.11: Hysteretic energy and maximum potential energy, for a single degree of freedom system.

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23

4 Eigen frequencies and fatigue damage

As eigen frequencies is a central topic in OWT analysis, a brief introduction of this topic will be given below. The aim is to give the reader a general background, as this is a central topic of discussion through the thesis.

Fatigue damage calculations has been used to quantify the effect of different soil-structure interaction models in the thesis. A brief theoretical presentation of fatigue damage calculation by S-N curves, will be given in this chapter. A more specific description of S-N curves and coefficients used in this thesis, will be given in Chapter 5.3.

4.1 Eigen frequency of OWT structures

A detailed analysis in the frequency domain is essential in OWT design. Both wind and wave loads appear cyclic on the wind turbine, possibly leading to resonance effects. Resonance effects fatigue life of the structure, and can influence optimal behavior for the OWT system. The frequency of some cyclic loads experienced by the structure is presented in Table 4.1.

Table 4.1: Frequency of cyclic loads on the NREL 5MW wind turbine

Frequency [Hz]

Period Wind - Synoptic peak (weather change)

- Turbulent peak

3x10^-6

~10^-2

4 days 90 s

Waves - Tidal waves

- Ordinary gravity waves peak

~10^-5

~0.1

12 h, 24 h 10 s

Rotor 1p - cut in

- rated

0.12 0.20

8.7 s 5.0 s

Rotor 3p - cut in

- rated

0.35 0.61

2.9 s 1.7 s

Generator - cut in

- rated

11.2 19.6

5.1x10^-2s 9.0x10^-2s

For OWT’s with monopile foundation, the 1p and 3p frequencies are often of highest significance, as they are in the region of the eigen frequencies of the structure. To avoid resonance, the system is designed to have eigen frequencies below, between or above 1p and 3p, as seen in Figure 4.1. Chapter 6.2 gives the lowest eigen frequencies of the 5MW NREL monopile, which is from 0.25-2 Hz. This is near the 1p and 3p frequency, and is therefore given extra attention.

Figure 4.1: 1p and 3p frequency relative to structure eigen frequencies.

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24 Finding eigen frequencies of the structure

The dynamic equation for an elastic system with linear damping can be expressed by the system equation of motion given by:

[𝑀]{𝑑̈} + [𝐶]{𝑑̇} + [𝐾]{𝑑} = {𝐹(𝑡)} (4.1) [M] is the system mass matrix, [C] is the system damping matrix, [K] is the system stiffness matrix, {d}

is the displacement vector of the system, and {𝐹(𝑡)} an external time varying force acting on the system.

For free undamped vibrations, the damping and external forces of the system is set to zero, giving:

[𝑀]{𝑑̈} + [𝐾]{𝑑} = {0} (4.2)

To find the natural frequencies of the system, the displacement vector is assumed to be according to:

{𝑑} = {𝜙}𝑒^𝑖𝜔𝑡= {𝜙}(cos(𝜔𝑡) + 𝑖 sin(𝜔𝑡)) (4.3) By derivation of equation (4.3), and substitution, equation (4.1) can be reduced to:

[[𝐾] − 𝜔²[𝑀]]{𝜙} = {0} (4.4)

This equation has a nontrivial solution only when |[𝐾] − 𝜔²[𝑀]| = 0. This system will have a number of solutions for the Eigen frequencies, 𝜔, equal to the dimensions of the mass and stiffness matrices.

This system can be solved in 3DFloat, giving the eigen frequencies of the subject system.

Soil-structure interaction modelling for an offshore wind turbine with monopile foundation